If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p2− q2 =
Given: a cot θ + b cosec θ = p
Squaring both sides, we get
(a cot θ + b cosec θ)2 = p2
⇒ a2 cot2 θ + b2 cosec2 θ + 2ab cot θ cosec θ = p2 ……(i)
and b cot θ + a cosec θ = q
Squaring both sides, we get
(b cot θ + a cosec θ)2 = q2
⇒ b2 cot2 θ + a2 cosec2 θ + 2ab cot θ cosec θ = q2 ……(ii)
To find: p2 – q2
Subtracting (ii) from (i), we get
a2 cot2 θ + b2 cosec2 θ + 2ab cot θ cosec θ – b2 cot2 θ – a2 cosec2 θ – 2ab cot θ cosec θ = p2 – q2
⇒ P2 – q2 = a2 (cot2 θ – cosec2 θ) + b2 (cosec2 θ – cot2 θ)
= a2 ( – 1) + b2 (1) [∵1 = cosec2 θ – cot2 θ]
= b2 – a2